Tire blew out.

So what was the explosion? Flat tire? Engine trouble? Grenade? Land mine?
"Nothing says I care like fatty meats." - Frasier Crane

Yep.
(As Henry goes on about getting the captains baggage out of the stolen vehicle)
Radar: Don't worry about the jeep, I'll change the numbers.
Henry: Oh, and change the numbers on that jeep.

Right. And there is some nice non sequitur humor re the jeep.
When Hawkeye arrives at the 4077
Col. Blake: It seems you stole a jeep up at headquarters.
Hawkeye : Oh-no-no sir. I didn't steal a jeep. It's right outside.
Col. Blake: So it is.
And when Hawkeye leaves in the jeep
Col. Blake: Did Hawkeye steal that jeep?
Radar : No, sir. That's the one he came in.
Col. Blake: Very good.

They are laughing as they drive away, as if they pulled off something together.
They're laughing that fate played in their favor.

How did Hawkeye and Duke know they wouldn't be chased by the MPs? They are laughing as they drive away, as if they pulled off something together.
"I will not go down in history as the greatest mass-murderer since Adolf Hitler!" - Merkin Muffley

If we could connect the dots, there might have been a scene where Hawkeye and Trapper put nails under the jeep's tires and KABOOM, the tire goes flat or they run over a small land mine. That's why the two of them are laughing as they drive away.
"I will not go down in history as the greatest mass-murderer since Adolf Hitler!" - Merkin Muffley

Not possible. Hawkeye arrives at HQ and waits beside the Jeep for his driver, then Duke arrives on the back of a truck. They obviously don't know each other, as Duke mistakes Hawkeye for the driver.

Well, it could have happened off-camera because it appears they are laughing about something sneaky, and they just met.
"It's like I'm talking to my Aunt Sylvia here!"

"The way it plays out makes it look like Hawkeye and Duke sabotaged the jeep but they did not, I guess."
Right, however it would be impossible for them to have done so.

The way it plays out makes it look like Hawkeye and Duke sabotaged the jeep but they did not, I guess.
"It's like I'm talking to my Aunt Sylvia here!"

It blew out a tire, and apparently the sergeant started badgering the two MPs and they just sort of got into it.