It's not clear to me, looking over all the various regulations and requirements, how (or even if) it is possible for students holding a four-year or five-year bachelor's degree from the US can work at a university as a part-time or full-time researcher. Is this possible? If so, are there specific programs to help set this up? Does it matter which EU country the institution is located? Is there a specific type of visa involved?
(Effectively, I guess this question could be boiled down to: what is the equivalent EU mechanism for a J-1 visa, when the student will be engaged in research rather than coursework?)
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.