Do I fold AA on this flop with this action?

[jman220]

So this was one of the longest amounts of time I've taken for a poker decision, I nearly ran the clock out. At first I completely felt that I was beat, and there was no way that button cold-calls an all in with anything less than a set. Then I realized that the pot was laying me 5:1 odds on the call, but I couldn't figure out if I"m really good 1 in 5 times. Then I realized that I also had outs, anywhere from 4-6, absolute worse case scenario being UTG playing AX clubs and Button having a set, leaving me with 3 clean straight outs, and 1 clean set out. I finally decided to call, but even after the hand I wasn't sure if it was correct (which is why I posted here). In fact I had all but convinced myself that I made a totally donk call. Consensus seems to be that it was the correct call however, so I feel better. Here's the hand history in full (I had to modify it slightly because converter isn't working properly):

Ultimate Bet No-Limit Hold'em, $2 BB (4 handed) converter

SB ($52.25)
Hero ($198)
UTG ($227.50)
Button ($390.85)

Preflop: Hero is BB with A[img]/images/graemlins/heart.gif[/img], A[img]/images/graemlins/diamond.gif[/img].
<font color="#CC3333">UTG raises to $7</font>, Button calls $7, <font color="#666666">1 fold</font>, <font color="#CC3333">Hero raises to $22</font>, UTG calls $15, Button calls $15.

Flop: ($67) 4[img]/images/graemlins/club.gif[/img], 5[img]/images/graemlins/club.gif[/img], 3[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(3 players)</font>
<font color="#CC3333">Hero bets $67</font>, <font color="#CC3333">UTG bets all in for $205</font>, <font color="#CC3333">Button calls</font>, <font color="#CC3333">Hero calls</font>.

Turn: ($654) A[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(3 players)</font>

River: ($654) 5[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(3 players)</font>

Final Pot: $654

Results (in white):

<font color="white"> Hero had AA
UTG Villain Had 10c/7c
Button Had 44 (for the set)

So I did indeed only have 4 outs, or maybe 4.25 if you count the ace of clubs plus the boat redraw as a quarter of an out. Not that the results matter.</font>