Can American students do research in Europe with only a bachelor's degree?

I think there are two parts in your question: can someone with a Bachelor from the US get a job from a university or research lab, and what kind of visa such a person would need. For the first question, it's really depending on the university. As for the visa, it depends if it would be for a PhD student, or for a "regular" researcher (i.e. not a student). However, in both cases, the visa application depends on the country, as far as I know, there is no Schengen Research Visa. For instance, for France, there are two types of Visa the person could apply to (Visa section of the french consulate in Washington)

• The “Skills and Talent” Card visa (Carte Compétences & Talents). However, for this one, they say:

  University graduates should be PhD. Lower graduates will be required to add proof of profesionnal experience in the same field for a minimum of 1 to 3 years (the lower the degree, the longer profesionnal experience expected)

• Visas for professors and researchers holders of a "convention d’accueil". According to this site (in French, I couldn't find an English version), a "convention d'accueil" would basically be the work contract with the university/lab, and the eligibility requirement seems to be a Master degree or equivalent (but it could be the case that a 5-years US Bachelor degree counts as an equivalent to a 5-year French Master degree). As far as I understand, that would be the best option for someone who wants to do her PhD in France.

It would also be possible to apply for a student visa, but I think it's not possible to get a work contract with such a visa. Another solution could be to apply to a:

• Employer Sponsored visa / OFII. There does not seem to be anything specific to research in this case.

So, to sum it up, I don't think there is a unified process to apply for a visa in Europe as a researcher, and each country has its own rules. Note that these visas are not automatically Schengen visa, and I've known the case of some extra-EU students, who obtained a visa in one Schengen country, and were not allowed to travel in other Schengen countries before becoming officially resident of the country.

It is possible to work as a researcher in a European University with only a bachelor's degree. In many fields, jobs as a research assistant would be pretty accessible to someone with this level of education, if your degree is in a relevant field or you have relevant experience. You might be able to get a slightly higher level of position if you have a good CV, and there is opportunity for advancement. However, ultimately it is very hard to advance into higher level academic positions without a PhD. This only happens in exceptional circumstances.

However, practically speaking this will be very difficult for someone without the right to work in the EU, such as a U.S. citizen. While getting such a job is technically possible (and varies by country), typically it involves the institution sponsoring you. This involves a lot of effort on their part, and most likely will only happen for higher-level positions. Also there may be barriers (such as salary level and education level requirements) that prevent you from getting a visa in certain countries, as well. For example, in the UK you would have to make at least £35,000 or have a PhD level job to get the relevant visa. In short, unless you have some unique, in-demand skills, this is unlikely to be a real option.

If you want to work as a researcher in Europe, consider studying for a higher degree in Europe. The barrier to entry is much lower for this; visas can be obtained relatively easily, and institutions provide a lot of support to help you. This need not be a PhD, you could just do a one year masters' degree. Depending on the country, you may be able to work while studying on a student visa. You also may have an easier path to transition to a work visa. In any case, the connections you make, and the additional education you get, should help.