According to most accounts it was dismounted men-at-arms who convinced the Swiss infantry to adopt pikes in the first place! At the Battle of Arbedo in 1422, the Milanese heavy cavalry dismounted and attacked on foot, using their long lances in both hands to outreach the swiss halberds and force them to flee. After the battle the swiss started to arm more and more of their infantry with pikes instead of halberds. During the 16th century, when men-at-arms did fight on foot it was typically as makeshift heavy pikemen up until lances fell out of use. At the Battle of Marigano in 1515, the French Gendarmes fought head to head against the Swiss pikemen either on foot or from the back of armored horses and eventually came out ahead.

As for how a pike square actually worked, the first thing to keep in mind is that until the very end of the 1500s a "pike square" wasn't made of pikes entirely. Generally it would look like a large square of pikemen surrounding a smaller square in the middle all armed with "short weapons"; i.e. halberds, greatswords, targets, etc. Aside from the standard bearers and their guards in the very middle of that square, the best armored and most experienced soldiers would all be placed in the outermost ranks and armed with pikes. The short weapons in the center generally had two roles. The first was to serve as a last resort in case the square fell apart and the battle devolved into a disorganized melee, the second was to pass through the ranks when the enemy started wavering in order to pursue and kill or execute downed enemies.

A two-handed thrust with a sturdy pike had a lot of mass behind. Even if it didn't pierce the weaker parts of an enemy's armor it typically had enough power to stop him in his tracks or knock him down, leaving the few who made it through to be quickly surrounded and cut down by swords, daggers, and halberds. On flat ground a well-ordered pike square was considered more or less invincible against everything but gunfire and other pikemen.

When two pike squares actually did come into contact it generally happened in one of two ways. The first was for both sides to slowly advance until the front ranks were close enough to "fence" with each other. This could go on for quite some time with individuals being replaced as they were injured or grew too tired to fight until one side had the upper hand.

The other method was for the pikemen to literally charge headlong into the enemy, making only a single, powerful thrust with their pikes before continuing forward with their swords. Blaise de Montluc described ordering his troops to do just this before the battle of Ceresole since he was concerned that the german infantry opposing him would be much more skilled at pike fencing than his Gascons. This tactic worked but with heavy casualties, as according to Montluc almost the entire front rank on each side was knocked down at the first impact and a large number of captains and other officers in the front ranks were killed or wounded.

So to sum up, men at arms certainly could serve very effectively on foot as heavy pikemen, though they might not be that much better armored than the front rankers of the pikemen opposite them. Even if they were on foot armed with pikes of their own though, they usually wouldn't be able to break a pike square in a single charge without taking heavy casualties of their own.