I don't have an Eggsbox so can't help your with your pit stops. Though there should be an option to setup the race strategy on the Engineers menu when in the garage just before the race.
In that Racer Strategy menu should be the option to set the number of pit stops and the tyres used.
Ignoring all that above, you can set your next tyre any time you are out on the track. You can pit at any time and change to those tyres.
Even when I set a pit strategy before the race on the PC, selecting my tyres, I still manually call them up during my in-lap. This is a habit I still have from the previous games in the series, where there used to be a bug where you might get the wrong tyres during the pit stop.
So no matter what tyres you have set in strategy, you can change that strategy on the fly. The tyres are on the DPAD menu, and when you press the DPAD (any part of the DPAD) the menu pops up. On that menu there is usually an option to change fuel settings, tyres, and brake balance. The example image below is not my image and the fuel option is missing from the menu (this a game option chosen elsewhere in the game settings).
In that menu you will see from the picture that DPAD-Right will bring up the
Tyre menu, and DPAD-Down will bring up the
brakes menu. When the tyre menu comes up, you will see an option to select Prime, Option, Inter, or Wet tyres. Choose your option (with DPAD up, down, left, or right), and the engineer will say something like 'OK'. Now when you enter the pits you will get the tyre you selected in that menu. You can do this at any time, but normally I do it on my in-lap or when I'm driving in the pit lane.
So the sequence might go like :-
DPAD-Right >>> Menu >>> DPAD-Right >>> Tyres >>> DPAD-Right >>> Options
or
DPAD-Right >>> Menu >>> DPAD-Right >>> Tyres >>> DPAD-Down >>> Wets