How can I filter an EntityClass by _not_ having a property?

You can use the operator UnequalTo for this purpose:

EntityList @ EntityClass["Country", "Continent"->UnequalTo["Europe"]]

{Entity["Country", "Afghanistan"], Entity["Country", "Algeria"], Entity["Country", "AmericanSamoa"], Entity["Country", "Angola"], Entity["Country", "Anguilla"], Entity["Country", "Antarctica"], Entity["Country", "AntiguaBarbuda"], Entity["Country", "Argentina"], Entity["Country", "Armenia"], Entity["Country", "Aruba"], Entity["Country", "Australia"], Entity["Country", "Azerbaijan"], Entity["Country", "Bahamas"], Entity["Country", "Bahrain"], Entity["Country", "Bangladesh"], Entity["Country", "Barbados"], Entity["Country", "Belize"], Entity["Country", "Benin"], Entity["Country", "Bermuda"], Entity["Country", "Bhutan"], Entity["Country", "Bolivia"], Entity["Country", "BonaireSintEustatiusAndSaba"], Entity["Country", "Botswana"], Entity["Country", "Brazil"], Entity["Country", "BritishVirginIslands"], Entity["Country", "Brunei"], Entity["Country", "BurkinaFaso"], Entity["Country", "Burundi"], Entity["Country", "Cambodia"], Entity["Country", "Cameroon"], Entity["Country", "Canada"], Entity["Country", "CapeVerde"], Entity["Country", "CaymanIslands"], Entity["Country", "CentralAfricanRepublic"], Entity["Country", "Chad"], Entity["Country", "Chile"], Entity["Country", "China"], Entity["Country", "ChristmasIsland"], Entity["Country", "CocosKeelingIslands"], Entity["Country", "Colombia"], Entity["Country", "Comoros"], Entity["Country", "CookIslands"], Entity["Country", "CostaRica"], Entity["Country", "Cuba"], Entity["Country", "Curacao"], Entity["Country", "Cyprus"], Entity["Country", "DemocraticRepublicCongo"], Entity["Country", "Djibouti"], Entity["Country", "Dominica"], Entity["Country", "DominicanRepublic"], Entity["Country", "EastTimor"], Entity["Country", "Ecuador"], Entity["Country", "Egypt"], Entity["Country", "ElSalvador"], Entity["Country", "EquatorialGuinea"], Entity["Country", "Eritrea"], Entity["Country", "Ethiopia"], Entity["Country", "FalklandIslands"], Entity["Country", "Fiji"], Entity["Country", "FrenchGuiana"], Entity["Country", "FrenchPolynesia"], Entity["Country", "Gabon"], Entity["Country", "Gambia"], Entity["Country", "GazaStrip"], Entity["Country", "Georgia"], Entity["Country", "Ghana"], Entity["Country", "Greenland"], Entity["Country", "Grenada"], Entity["Country", "Guadeloupe"], Entity["Country", "Guam"], Entity["Country", "Guatemala"], Entity["Country", "Guinea"], Entity["Country", "GuineaBissau"], Entity["Country", "Guyana"], Entity["Country", "Haiti"], Entity["Country", "Honduras"], Entity["Country", "HongKong"], Entity["Country", "India"], Entity["Country", "Indonesia"], Entity["Country", "Iran"], Entity["Country", "Iraq"], Entity["Country", "Israel"], Entity["Country", "IvoryCoast"], Entity["Country", "Jamaica"], Entity["Country", "Japan"], Entity["Country", "Jordan"], Entity["Country", "Kazakhstan"], Entity["Country", "Kenya"], Entity["Country", "Kiribati"], Entity["Country", "Kuwait"], Entity["Country", "Kyrgyzstan"], Entity["Country", "Laos"], Entity["Country", "Lebanon"], Entity["Country", "Lesotho"], Entity["Country", "Liberia"], Entity["Country", "Libya"], Entity["Country", "Macau"], Entity["Country", "Madagascar"], Entity["Country", "Malawi"], Entity["Country", "Malaysia"], Entity["Country", "Maldives"], Entity["Country", "Mali"], Entity["Country", "MarshallIslands"], Entity["Country", "Martinique"], Entity["Country", "Mauritania"], Entity["Country", "Mauritius"], Entity["Country", "Mayotte"], Entity["Country", "Mexico"], Entity["Country", "Micronesia"], Entity["Country", "Mongolia"], Entity["Country", "Montserrat"], Entity["Country", "Morocco"], Entity["Country", "Mozambique"], Entity["Country", "Myanmar"], Entity["Country", "Namibia"], Entity["Country", "Nauru"], Entity["Country", "Nepal"], Entity["Country", "NewCaledonia"], Entity["Country", "NewZealand"], Entity["Country", "Nicaragua"], Entity["Country", "Niger"], Entity["Country", "Nigeria"], Entity["Country", "Niue"], Entity["Country", "NorfolkIsland"], Entity["Country", "NorthernMarianaIslands"], Entity["Country", "NorthKorea"], Entity["Country", "Oman"], Entity["Country", "Pakistan"], Entity["Country", "Palau"], Entity["Country", "Panama"], Entity["Country", "PapuaNewGuinea"], Entity["Country", "Paraguay"], Entity["Country", "Peru"], Entity["Country", "Philippines"], Entity["Country", "PitcairnIslands"], Entity["Country", "PuertoRico"], Entity["Country", "Qatar"], Entity["Country", "RepublicCongo"], Entity["Country", "Reunion"], Entity["Country", "Russia"], Entity["Country", "Rwanda"], Entity["Country", "SaintBarthelemy"], Entity["Country", "SaintHelena"], Entity["Country", "SaintKittsNevis"], Entity["Country", "SaintLucia"], Entity["Country", "SaintMartin"], Entity["Country", "SaintPierreMiquelon"], Entity["Country", "SaintVincentGrenadines"], Entity["Country", "Samoa"], Entity["Country", "SaoTomePrincipe"], Entity["Country", "SaudiArabia"], Entity["Country", "Senegal"], Entity["Country", "Seychelles"], Entity["Country", "SierraLeone"], Entity["Country", "Singapore"], Entity["Country", "SintMaarten"], Entity["Country", "SolomonIslands"], Entity["Country", "Somalia"], Entity["Country", "SouthAfrica"], Entity["Country", "SouthKorea"], Entity["Country", "SouthSudan"], Entity["Country", "SriLanka"], Entity["Country", "Sudan"], Entity["Country", "Suriname"], Entity["Country", "Swaziland"], Entity["Country", "Syria"], Entity["Country", "Taiwan"], Entity["Country", "Tajikistan"], Entity["Country", "Tanzania"], Entity["Country", "Thailand"], Entity["Country", "Togo"], Entity["Country", "Tokelau"], Entity["Country", "Tonga"], Entity["Country", "TrinidadTobago"], Entity["Country", "Tunisia"], Entity["Country", "Turkey"], Entity["Country", "Turkmenistan"], Entity["Country", "TurksCaicosIslands"], Entity["Country", "Tuvalu"], Entity["Country", "Uganda"], Entity["Country", "UnitedArabEmirates"], Entity["Country", "UnitedStates"], Entity["Country", "UnitedStatesVirginIslands"], Entity["Country", "Uruguay"], Entity["Country", "Uzbekistan"], Entity["Country", "Vanuatu"], Entity["Country", "Venezuela"], Entity["Country", "Vietnam"], Entity["Country", "WallisFutuna"], Entity["Country", "WestBank"], Entity["Country", "WesternSahara"], Entity["Country", "Yemen"], Entity["Country", "Zambia"], Entity["Country", "Zimbabwe"]}

Addendum

For excluding multiple value, you could use the following syntax:

EntityList @ EntityClass["Country",
    {
    "Continent"->UnequalTo["Europe"],
    "Continent"->UnequalTo["Asia"]
    }
] //Length

139


Alternative solution with FilteredEntityClass and EntityFunction

The code

For the sake of documenting this one more time on this site, a more general answer (w.r.t. the answer of Carl) would be to use EntityFunction and FilteredEntityClass. In this particular case, one additional subtlety is that in this method, one has to compare the "Continent" property with EntityClass["Country", "Europe"], rather than a string "Europe".

Here is the code

nonEuropeanCountries = EntityList @ FilteredEntityClass[
  "Country",
   EntityFunction[
     c, 
     c["Continent"] =!=  EntityClass["Country", "Europe"]
   ]
]

Differences with the operator-based method

Note that this result will be different from that obtained with Carl's code, since this one keeps the countries for which the continent is not defined, while Carl's code throws them away:

Complement[
  nonEuropeanCountries, 
  EntityList @ EntityClass["Country", "Continent" -> UnequalTo["Europe"]]
]

(*
{
  Entity["Country", "BouvetIsland"], 
  Entity["Country", "BritishIndianOceanTerritory"], 
  Entity["Country", "FrenchSouthernAndAntarcticLands"], 
  Entity["Country", "SouthGeorgiaAndTheSouthSandwichIslands"], 
  Entity["Country", "UnitedStatesMinorOutlyingIslands"]
}
*)

Which behavior is more desired, depends on the situation, but it is important to keep in mind this difference. Also, while one can filter such countries out from the above solution, one can't reconstruct those back from the one based on operators.

More on EntityFunction

In general, EntityFunction is a somewhat more verbose, but also significantly more powerful device, than the short-hand syntax based on rules, since one can create predicates which couple together more than one entity property - which is in general impossible in the short syntax (except cases when the full predicate can be expressed via individual property predicates combined using simple And condition).

Here is an example that would be pretty hard to express in the short syntax, within a single query: find all countries which either are in Europe and their names start with "A", or have population higher than 100 million people. Here is the code for this using EntityFunction:

EntityList @ FilteredEntityClass[
  "Country",
  EntityFunction[c, 
    Or[
      And[
        c["Continent"] ===  EntityClass["Country", "Europe"], 
        StringStartsQ[c["Name"], "A"]
      ],
      c["Population"] > Quantity[10^8, "People"]
    ]
  ]
]

(*

{
  Entity["Country", "AlandIslands"], Entity["Country", "Albania"], 
  Entity["Country", "Andorra"], Entity["Country", "Austria"], 
  Entity["Country", "Bangladesh"], Entity["Country", "Brazil"], 
  Entity["Country", "China"], Entity["Country", "Ethiopia"], 
  Entity["Country", "India"], Entity["Country", "Indonesia"], 
  Entity["Country", "Japan"], Entity["Country", "Mexico"], 
  Entity["Country", "Nigeria"], Entity["Country", "Pakistan"], 
  Entity["Country", "Philippines"], Entity["Country", "Russia"], 
  Entity["Country", "UnitedStates"]
}
*)