Can the company tell her that I can't go with her?
Probably, although possibly not. If the picnic is on company property or at some privately owned facility then they can exclude you if they want. If it is at a public park or other sort of public facility, it is possible that they can't forbid your attendance; it would depend on laws and contracts, which are outside the scope of this site. However, if the company doesn't want you there it would be best to stay away.
Note however the "if they want" above. What you haven't said is if they have told you (or your daughter) that they want to you stay away. Have you asked? Under the circumstances, it's possible that no one would mind, unless there are bitter feelings held against you by management and/or your former co-workers. From what you've said, that doesn't seem to be the case here. So, ask if it's okay for you to attend, or - maybe even better - have your daughter ask.
Partly this is based on a somewhat similar personal experience long ago, except I resigned, instead of being fired. The call to request tickets for the company picnic went out about six weeks before the picnic. Not knowing for certain if I would still be employed by them, I requested a ticket, even though I was job hunting. In that six week I was offered a new job and accepted it. My last day on the job was a Friday and the picnic was 2 days later, on Sunday. I asked around some and no one thought I should avoid the picnic, so I attended. My line manager gave me some good natured ribbing when I was at the picnic, but no one seriously objected and we had a good time.