I have a disconnect box that is on a 50 amp breaker connected to a disconnect box using 6 AWG. In order to connect my mini split system to the disconnect do I use 10 AWG wire based on the information on the label?
You can use any wire you want, 10 AWG or larger... so if you have some #6 lying around, don't make a run to the store lol.
As far as circuit breaker, 30A is fine. What the label is telling you is that if you get nuisance trips, it's OK to bump to 35A or 40A. Or even 45A, but that is not a commonly made breaker size, so that basically means 40A lol. It might be available in fuses.