Yes provided your AR 10 uses the standardized trigger pin size and pattern BUT there is a slight risk of getting light primer strikes on heavy primer surplus 308. Triggertech does make version for ar-10 applications with a harder hitting hammer to eliminate this issue.
While the Triggertech AR-15 Two Stage Duty Trigger was not specifically designed for .22LR, it can be used with an M&P 15-22 rifle.