You can only say if the person has worked there anyways. You can't ask why they are no longer employed there.
Lots of misnomers in this thread, like this one. A past employer is legally entitled to give a candid opinion of you if contacted by a prospective employer. So yes, if you had problems being punctual, were caught stealing, got caught coming into work drunk or high, spread "office" gossip, got fired for sexual harassment... or heck, even if you left on good terms but your boss thought you were a lousy employee-even if such thoughts were unjustified... UPS has a legal right to say this. But here's the problem: if that prospective employer declines your employment... you have a legal right to know if it was in part because UPS gave a poor recommendation. Because of this, many large companies have a policy in which they will only confirm a person's employment, and provide no other details. That use to be (and may still be) UPS's policy.
I don't recall seeing such a question on the employment application. "Do you currently work for FedEx?"
If you're referring to "employment history"- I've always taken that to mean... "places you've worked at and wouldn't mind us calling". You typically leave off any any places where you were fired from, or in this case...
That employment application asks you if you've provided accurate information. This is stuff that typically gets people into trouble.
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An employer is legally entitled to a full effort from you. If it feels another job is preventing you from giving it 100% of your effort... it can legally let you go/ask you to pick. Heck, if it feels that 60 minutes you spend at the gym before coming into work is preventing you from giving 100%... it can legally let you go/ask you to pick. Obviously with the union, this is problematic at UPS but I have seen UPS do this once (in my 15 years): a PTer worked a 12-hour shift before coming into UPS. His punctuality was an issue as was his performance... hindered by the fact that he struggled to stay awake. If he was scanning, he'd miss a ton of scans. He was only at UPS because of the benefits/opportunity to go FT and thought because of the union, UPS couldn't do anything to him. He thought wrong.
Most employers don't enforce the above, but they do have "conflict of interest" policies for obvious reasons (well, maybe not to some BC members). To answer the OP's question, if you're a PT hourly, UPS will generally look the other way if you're also working for FedEx. But if you move into a consumer facing position (drive, consumer counter, etc.) OR a managerial position (even PT sup), you will not be permitted to work for the competition. FedEx, OTOH, may have a problem w/it...