In a unionized workforce, the ability to create/enforce rules, including work rules, depends on the language of the collective bargaining agreement and past practice.
If the NFL has any duty (arising from either the collective bargaining agreement or the parties past practice) to bargain over a rule, they may NOT have the right to "make its own rules and enforce them" - they may have given up that right.
Game rules - MUCH more likely the answer is yes,
Conduct rules - possible they gave their unilateral rights away in bargaining.
Same type of issue will impact (but may not completely control) Goat's point about "spoliation of evidence" (destroying the phone). In a regular lawsuit, Goat is right - once you believ there might be a lawsuit, you (and your lawyers) have an affirmative duty to to take affirmative steps to preserve evidence, including electronic evidence. (The battle over preservation of electronic evidence subsumes many lawsuits these days.) But a labor law dispute is much different, and this ultimate case will likely wind up in arbitration, not court.
There will be a quick "injunction" hearing and a judge will decide if Brady can play while the case proceeds (I suspect he will play), but the ultimate decision (can NFL/Goodell do THIS in THIS WAY) requires interpretation of a collective bargaining agreement, and judges don't do that. That is for arbitrators. And that process won't be quick (unless both sides put their foot on the gas and try to get done quick. But the union can easily drag this out until after this season.)
Lots of issues here, and process matters as much/more than substance.
NFL rights.
Team rights.
Brady rights.
Union rights.
All of which are slightly different.