According to the regimental history by major general Sir Frederick Maurice (History of the Scots Guards) the regiment originates from a regiment founded in 1642 for the Scottish expedition to Ireland. This regiment, raised by the Archibald, 1st Marquis of Argyll (the 8th Earl of Argyll and father of the 9th Earl; the latter became known for his revolt against James II in 1685. Both shared the unfortunate fate of being executed.), would be formed into a regiment of foot guards after its return from Ireland in 1649. As such it served Charles II at Worcester.
However, a nice article titled The Myth of the Founding of the Scots Guards in 1642 by David Stevenson (appeared in The Scottish Historical Review, Vol. 56, No. 161, Part 1 (April 1977)) puts this old origin into perspective.
One main error made by Maurice, according to Stevenson, is that Argyll (i.e. the Campbells) was in fact hostile to Charles I and the Scots privy council. Thus it seems unlikely that Charles I would commission Argyll to raised a regiment of guards for his protection! That Argyll received a commission is beyond doubt, but his regiment was just as any other of the Scottish regiments intended for Ireland.
The second misunderstanding relates to the supposed transformation of the 1642 regiment into a lifeguard of foot in 1650 (after its return from Ireland). As Stevenson argues, this lifeguard was indeed formed in 1649 as Irish companies and designed to be a lifeguard of foot in 1650. However, this new regiment did not descend from the Argyll regiment of 1642. Though few men may have served in both regiments, the 1649 Irish Companies were formed from the many Scots that had fled from Ireland in 1648 and 1649.
Furthermore, Stevenson states that
the guard was intended to be formed not of men loyal to the king but of men the suspicious kirk party could trust not to sympathise with Charles II
Thus, this 1650 lifeguard of foot had everything to do with internal Scots politics, and much less with loyalty to and love for the person of Charles II. It is unlikely that the foot guards raised in the early 1660s are to be considered successor to the lifeguard of foot of 1650.