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 &amp; 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.