Can I mention University B's name in an supplemental essay that I am writing for University A?

I would say yes... as long as you emplace your enthusiasm toward the school you are applying for

If you took a summer program at another university and wish to ONLY imply the significance of the event (not the school that held it), I believe it would be okay to detail your experiences within the event and showcased your growth in your essay. But ask yourself, “how significant was this program to me and my goals?” Was it an internship where you worked alongside a postdoc, or was it something you applied for to earn? Either one exhibits your initiative to take on a new challenge or opportunity, but it’s all about how you present that.

Furthermore, everything your portray in your essay should be highly reflective of the prompt. In fact, what type of prompt was it for? If you included that I’d be able to answer better, but I don’t think it will matter much in any essay. For example, mention the program once in full and then abbreviated throughout the essay: “I attended a Clinical Shadowing program at University of State (CSUS)....while at CSUS, I learned.... I plan to apply the experiences I’ve gained from CSUS to [desired/applying university] to earn....” These are all hypothetical, contextual examples, but I suggest following the syntax and the setup.

A bad example: “The University of State held an extraordinary program over the summer—Clinical Shadowing—and I’ve learned a great deal about.... while at the University of State, I experienced.... I plan to input what I’ve learned from the University of State to [desired/applying university] to earn....” Another hypothetical example, but this would imply too much emphasis on the undesired school. I suggest you use the former example.

Hope this put into perspective the value of writing; in essence, you can shift and shape anything you feel uncertain about into a essay that portrays the message effectively. In your case, it’s all about EMPHASIS, which college needs to stand out more in your essay

It's not a problem at all, but I would ask how necessary you think the college and program name really are for your essay. If you're trying to name drop for prestige, then the programs that that would be worth it for are probably not all that plentiful, as mentioned below.

If you're not doing it to identify the program's prestige, but just because it's awkward logistically not to include the name of the program, obviously it's hard to get around that. You just want to be very careful that your mention of University B is essentially just for that logistical reason: you don't really want your essay and the reasons for why the program at University B was meaningful to you revolve around characteristics specific to University B.

Which also leads me wonder a little bit about your prompt: it sounds like this is maybe a why major type of prompt, or something of that sort. You want to ask yourself whether it's really imperative that you refer to this other program to make your point about University A, or if that space would be better served if used to describe University A more in depth and your specific fit to University A's relevant programs/majors, not your fit to University B's programs that in your opinion have parallels to University A's programs. I don't know the specifics of your prompt or your essay, so I obviously can't make that choice for you. But just keep in mind that even if your experience at University B's program is a really meaningful, interesting topic to write about, a lot of writing for college prompts involves making hard choices! There might be better, more effective ways to make your point (and there might not! again, that's something only you can really decide).

So it is absolutely okay just dont say I am so eager to attend Univ B in an essay you are submitting to Univ A. For example, I attended a seminar/workshop for one of my top majors but I have since lowered my ranking on that school (you probably heard of it lets say State St) Id mention that as an activity I did. You can talk about the program and you can name drop the university to show the prestige of the university. The only exception is an uncompetitive program that costs a lot to attend then its not worth doing an essay about it.

Id talk about what I learn at State St 3 day seminar of insert major but of course focus on the program not the university but naming the university is perfectly acceptable.

Hope this helps and feel free to comment if you need something clarified.