I second Flipflopsalto's advice to treat them more like adults. Sticker/award charts and such reinforce the model that they are perceived as little kids and the standards for their behavior are consequently lower. Treating them like adults sets the expectation that you expect behavior approaching the maturity of adults.
If you had an adult class, and people were being too chatty while you were giving instructions, or taking breaks to talk on their phones, what would you do?
Remind everyone that it's important for each person to be able to hear instructions and your corrections, so keep the chattiness down. It's also disrespectful to be too chatty while in any class environment. Give reminders for slips, but if it continues, call out the chatternbox and talk to them 1 on 1 about their desire to learn gymnastics, and how you can't help them if they're not paying attention or distracting others around them. If it still doesn't stop, involve the parents and discuss the same and how his/her chattiness is affecting your teaching.
For phones/breaks/bathroom , again, just explain the rules. If a kid is avoiding class time with excess breaks, again talk to him/her 1 on 1 and inquire about their desire to gain skills and be in gymnastics and how you can't help them if they're taking undue breaks. If it continues, again, time to involve the parents.
Above all, make it clear to all that if the pattern of disrespect and not paying attention continues, they may not be invited to continue in the class, or at the very least, you will only be focusing your time on the students who DO show you that they care enough to present and attentive. The others will not be invited to try new skills and will not make much progress as a result.
My 2c as a parent!
Good luck!