"Squires and knight in another knights retuines" Topic

This is during the high medieval period with coat of arms on surcoats. Would squires and lesser knights that are part of a household/retinue of a more important knight wear his coat of arms?

No! As knights they would wear their own arms. The only people who would wear a nobles arms are himself and possibly his 'herald' – everyone else would wear badges or livery if they were 'his men'.

Practices probably varied. As said, a knight would normally bear his own arms even if in another man's retinue. A non-knighted man like an esquire might have done the same if he came from an armigerous family – the right to bear arms (heraldic arms, not the other sort) is now, and I think always was, based on descent not knighthood.

With lesser men it is less certain. Sometimes they would indeed display their lords' arms. For instance Froissart says: "He was accompanied by the constable of Aquitaine, sir John Chandos, who had under him full twelve hundred pennons, all ornamented with his arms, which were a sharp pile gules, on a field argent" – but he doesn't say what the bearers of those pennons bore on their shields. We also have Joinville on the Count of Jaffa in the 13th century "He it was who made the most noble show at landing; for his galley came up all painted above and below water with his escutcheons, the arms of which are "or with a cross gules patee." He had about three hundred oarsmen in his galley, and each oarsman bore a target with his arms, and to each target was attached a streamer with his arms embossed in gold".

Hm, interesting.
What about sons of the knight who are either a knights or their own of squires?

That may depend on date, and certainly depends on the country. In more recent heraldry, in England, Scotland and France at least, there are fairly rigid systems of "marks of cadency" where sons "difference" the father's arms with a mark that differentiates their arms from their father's and those of other family members. In the modern English system, the first son bears a label, the second a crescent, the third a mullet ( star ) and so on. I think this developed gradually around the 14th century; earlier, there was some use of labels and so on but it was less systematic. ( I am very much open to correction on chronology. )

For a couple of examples, see the Caerlaverock Roll ( link ) – "Robert le FitzRoger's banner … quartered was of gold and scarlet, over all a bend of sable; John, his son, de Clavering surnamed, bore the same with a green label".

Some other countries and regions do not seem to have used much differentiation within the family at all.