This proposition has basically two parts: it proposes that The Red Crosse Knight is an "Everyman" figure and that he is shown to have achieved his quest by the end of Book I. So, I'll take each on each part separately, though the easy answer to each is that it all depends on one's point of view.
Whether or not Red Crosse is an "Everyman" depends on one's point of view to the extent that he seems not to have a unique or well-defined identity in Book I and that he has a name that is simply descriptive of his armor, the same as if one were writing about "The Black Knight" or "The Red Knight" or "The White Knight," etc. His character could be said to be a bit more defined to the extent that the red cross on his armor and shield are associated with Christian iconography, but on the other hand we are told in the first Canto that this armor is borrowed-- it's not even his own! It is marked with dents and scars from knightly fights in the past, but this knight did not fight them. Now, we could assume that this knight aspires to become the knight who had inhabited that armor, but that is only an assumption, isn't it?
Book I is described as the story of the "Knight of Holinesse," but unlike the other Books of the FQ, this knight has to become holy, has to come to learn his identity during the quest. For example, in Book II, Sir Guyon IS the Knight of Temperance; he has a name, and he possesses the quality of temperance long before the action of Book II opens-- the quest has to do with his teaching others his quality or ridding the world of intemperance.
But this is not so with Book I. Red Crosse, wearing borrowed armor, sets off on his quest to free Una's parents from the dragon, and the first thing he does is to become lost in the Wood of Errour. This element of the plot allegorically points to the fact of his quest for his identity, since a true Knight of Holiness would not so easily lose himself (and his Lady) in Error. We could also discuss what he encounters there-- the monstrous Errour, herself, who nearly subdues RC merely with slime and stink! He has a LONG way to go before he is ready to fight a dragon! So, it's (I hope) a little clearer to see that RC is not like the others of Spenser's Knights of each Book of the FQ. He has a double quest, it seems: he must first achieve his identity before he can even hope to have a chance against the dragon who holds Una's parents in thrall.
To that extent, then, we could certainly argue that RC is an "Eveeryman" figure-- more humanized, he does not know himself; he begins in weakness and must build up to strength, lilke the rest of us. He doesn't even know his name until he reaches the Mount of Contemplation and is shown who he is in a supreme vision of the future and of his place in it.
I will continue my answer in another post, as this one is drawing near my word-limit!