Hello there !

First thing first, I've never heard such a "fact", or seen any reference to it. That may be an idea that originate from one particular school system, I don't know, but I think it is important to point out that it is not necessarily a widespread theory.

Now to the question. As you mentionned it, "medieval times" are a period that lasts approximately a thousand years (depending on the starting and finishing years you pick). As you guessed it, things vary a lot through time. Therefore, I won't be able to provide an extensive answer to your question for the entirety of the time period. I'll keep to my specialty, that is XIV-XV^(th) century, particularly France.

On that period, the idea of a defined schedule "page-squire-knight", all lasting 7 years, is not true. In order to explain why, we need to talk a bit about what being a squire or a knight meant.

In early middle ages, a squire was expected to become a knight eventually. A squire basically was a "knight in being", or in training. He was following a knight that provided him some kind of formation and protection, until the day he was deemed worthy to become himself a fully independent knight. Until then, he could be considered of secondary value on the battlefield, if he even fought. He could very well be left behind in the camp. That is also due to the fact that as squires all became knights, they tended to be younger.

In late middle ages, however, that was not the case anymore. Squires were not only considered knights to be anymore, or at least not only. It was a mandatory step to becoming a knight, yet knighthood was not granted to every squire. With time, knighthood became more of a source of prestige and financial gain (we'll see about that later) than the fulfillment of any nobleman training.

Squires became valuables assets on the battlefield and, in the end, the only thing that separated a squire from a knight was a ceremony and some kind of social recognition. Yet a squire was heavily armored, well trained, etc. He was part of heavy cavalry (or sometimes infantry) just like any other knight. That is also due to the fact that, since squires did not become knights automatically, the were older than in early middle ages.

The main difference, if we ignore the social standing and prestige that being a knight brought, is on the matter of money. French nobles were expected to provide for 40 days of military service a year to the king. That is extremely few, considering some of them had to journey for several weeks to come from their lands to the rendezvous point of the ost (royal army). Any exceeding day was to be paid, and a distinction was made : you were paid according to your rank : banner lord, knight, squire or commoner. Yet the pay for military service was not excellent and did not make for a great part of the money a noble could make in war (as opposed to ransoms, for instance).

Knighthood became more and more of a prestige thing with time, losing gradually its military elite connotation. That was already nearly the case by the early XVth century, and became truer by the day during the late XVth century.

There is some king of paradox to the affirmation of 7-14-21. If you want to apply it to early middle ages, where squires were indeed all becoming knights, it makes little sense. Knighthood was the ultimate step of formation and training, and made you part of a military elite. Then there is little interest in setting an age limit : you're trained and ready when you are, not when the timer says so.

If you want to apply it to late middle ages, you can see it does not work either, since many a squire never became a knight.

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I hope this helped answering your question !