Okay, so step one to fighting the boss is unlocking him to fight against, this can be done by completing planets in the Bugs system. Completing planets award influence based on how many stars you gain in missions and the difficulty level of the planet. Once you have 50 influence you can initiate the boss fight.
This fight is probably the most tricky of the three, at least to solo as I tend not to boss fight with a group. This boss is very mobile and doesn't allow for much sustained damage. It burrows underground and only pops up to spit out some acidic damage around the battlefield or melee attack a player. These moments are your only opportunities to deal damage do you must be prepared to strike once he comes above ground
The method I've currently found to work best is a bunch of fully upgraded EAT-17s and the Stratagem Priority perk. Spamming these all over the place, picking them up and firing them at the boss whenever possible, although VERY slow seems to be the most reliable and easy way I can find to deal with the boss in my little experience with it.
An alternative with a full team is to run Mechs. Why? Well, while you only get a few rockets per Mech, a fully upgraded Mech can do significant damage by firing all its rockets thanks to its anti armour nature, having 4 Mechs each between 3 or 4 players is more than enough to win the fight with ease, the only problem is coordinating when and where to drop your Mechs in the fight as the bug boss is very mobile and will target loose Mechs, destroying and wasting one of your Mechs.
Similarly to the other boss solutions I made, if I record a video demonstrating either method I will edit the solution with it.