Chain Challenge and Squad Assault are both intended for advanced players and require you to take on extremely powerful foes in a series of multiple battles. You'll need to form several teams, meaning you'll need to be on top of developing your Heroes and building up your force's overall strength.
You can earn Orbs as rewards for your first clears in either mode (in the case of Chain Challenge, when you take on Lunatic difficulty). These modes may be difficult, but the Orbs you earn can help you build out even more powerful teams, so be sure to use this opportunity to stock up!
In Chain Challenge, you'll generally have to take on a series of five or ten battles taken from the Main Story, or three or six battles taken from Paralogues. Your allies' starting positions will be the same as in the original maps, so this might help you to hone your strategies. You'll receive Hero Feathers as rewards for clearing Normal and Hard, and Orbs for clearing Lunatic.
You'll be able to continue on to the next map with your current team for as long as it remains undefeated. If all members of the team are defeated, you'll need to form a new team from your remaining Heroes. You'll still be able to clear the challenge as long as you remain within the permitted number of teams and have at least one survivor in the end.
Squad Assault tests your skills by applying some restrictions. For example, you'll need to form teams for each map, and once you've deployed a Hero, you won't be able to put them in another team. Unlike Chain Challenge, you'll need to have all four members of the team survive, making your task extremely difficult. Rewards from this mode include Sacred Seals and Orbs.
You'll need to form teams of unique Heroes for five maps to be fought in a row, meaning you'll need 20 Heroes total (4 Heroes x 5 maps). Once you've deployed a Hero, you won't be able to use them in subsequent battles, and you can't have multiples of the same Hero on a team.