Assign multiple values to enum elements

You will have to do something like this:

class Region
{
    static readonly RegionMap = new Dictionary<int,string>
    {
        { 1, "US" },
        { 2, "US" },
        { 3, "Canada" }
        { 4, "Canada" }
    }

    public static string GetRegion(int code)
    {
        string name;
        if (!RegionMap.TryGetValue(code, out name)
        {
            // Error handling here
        }
        return name;
    }
}

Then look up the string based on the value from the database:

string region = Region.GetRegion(dbVal);

Maybe something like this?

static Country ConvertRegionToCountry(string region)
{
    switch (region)
    {
        case "1":
        case "2":
            return Country.US;

        case "3":
        case "4":
            return Country.Canada;
    }
}

This is not possible. You'd have to use seperate values then. If the names are the same, ie.

[Serializable]
[Flags]
public enum Country
{
    US      = 1,
    Canada  = 2,
    Northern = 4,
    Southern = 8
}

You could do this: Countries = Country.US | Country.Northern. If not, you need to find another way, possible another property or even better, a Location class.

An enum may not be the right construct to model this kind of problem.

I would suggest creating a class to represent country information, and provide methods the convert to and from numeric representations. With problems like this, you also have to decide what coding value you will use when converting a selected Country instance into a numeric value.

The Enum Object pattern can be helpful starting point for modeling this kind of situation:

public sealed class Country
{
    // initialize appropriately in the constructor...
    private readonly int[] m_Values;
    private readonly string m_Name;

    // make the constructor private so that only this class can set up instances
    private Country( string name, int[] codes ) { ... }

    public static Country US = new Country("United States", new[]{ 1,2 } );
    public static Country Canada = new Country("Canada", new[] {3,4} );

    public static Country FromCode( int code ) { ... }
    public override string ToString() { return m_Name; }
    // ... etc...
}

Based on your example, you should also consider whether you need to model Country sub-regions as first-class entities, rather than simply folding them into the implementation details of your Country enumeration. Whether you should do this or not depends on your requirements and use cases, so only you can make an appropriate decision on that.