if they focus to a red boost can they get an additional action before the boost? And if so, do they end up with 2 stress afterwards?

You can do a linked action off the action you get from Pattern Analyzer, if you want to end up double stressed.

(or, if you're Nien Nunb, you can do it and get rid of both stress, then take your perform action step as normal.)

Not sure what you mean by 'an additional action before the boost' though.

Pattern Analyzer does not trigger off of a boost, because a boost is not a maneuver.

If you fully execute a red maneuver and choose to use Pattern Analyzer, then yes you can use your Refined Gyrostabilizers ability to take a boost or turret rotate action, after which you will be double-stressed.

Note that Check Difficulty comes after laying down the template and moving the ship.