Not overly familiar with that particular radio but if the 777 is like any other "ham radio" out there, you should have the option to
independently program CTCSS tones on the transmit as well as the receive frequency. In other words, you could select CTCSS on transmit to unmute your team radios but have carrier-squelch only on receive so you hear everything regardless. Or you could do the reverse but that would be extremely counter-productive as you'd hear your team because you have CTCSS receive enabled, but they wouldn't hear you because you're not transmitting CTCSS yourself. Usually, you either program CTCSS on both tx/rx, or you have CTCSS tx-only and use carrier squelch on the receive.
However with FRS/GMRS, there's usually enough discrete channels available that rather than using CTCSS or another form of conditional access squelch system (DTMF burst, DCS, etc.), most teams/events just pick a channel that the other isn't using and run with it. Less complicated that way than having to fumble with the correct squelch codes.
Fly