Is there any significance to the use of "When Johnny Comes Marching Home"?

First of all, as Oliver_C points out in his comment,

The first movie used Beethoven's 9th Symphony, the second Jean Sibelius' Finlandia, so it seems like a tradition to pick another "old" musical piece as sort of a leitmotif for the third movie (btw, all three movies were scored by the same composer: Michael Kamen)

And especially this song with its style of a military march (and the extensive use of drums throughout the movie) fits well to the military background and demeanor of the bad guys, who are more than one time depicted as being (or at least seeing themselves as) soldiers and behaving accordingly:

Simon: I'm a soldier, not a monster. Even though I sometimes work for monsters.

Add to this that the song is basically about returning from war, and even more so the more pessimistic version of Johnny I Hardly Knew Ye given in dglickler's answer. So it's motifs again fit to the bad guys' background of being ex-soldiers. Seeing that they are depicted as former GDR soldiers and the movie is likely set around its release in 1995, they might feel like having been taken away their homeland and with that basically their existence as soldiers. So they more or less feel a bit like disillusioned veterans returning from war (albeit a rather cold one), not knowing what to do now or what to fight for and trying to get their share from the world. This is also emphasized in Simon's speech at the end:

Yesterday we were an army with no country. Tomorrow we have to decide which country we want to buy!

There's two real links.

First: "When Johnny Comes Marching Home" is also known by another name: "Johnny I Hardly Knew Ye." In both versions of the song, more the second, Johnny is a man who has returned from war...missing many pieces. The song strips away piece by piece of his body, leaving him a lump. The one who he comes back to is left angry, resentful, and hurtful. "Ye haven't an arm and ye haven't a leg, haroo, haroo."

The tie-in there to Die Hard With A Vengeance relates partly to the way the enemy (Simon) picks away at him to try to win (the guards, the false leads, etc.) and partly to the fact that Simon is angry, vengeful over his loss of his brother.

The interesting thing is that this song was left off of the soundtrack.

For interest, the lyrics of "Johnny I hardly knew ye" are:

When on the road to sweet Athy,
Hurroo Hurroo
When on the road to sweet Athy,
Hurroo Hurroo
When on the road to sweet Athy
A stick in the hand, A drop in the eye
A doleful damsel I heard cry
Johnny I hardly knew ya

Where are the eyes that looked so mild,
Hurroo Hurroo
Where are the eyes that looked so mild,
Hurroo Hurroo
Where are the eyes that looked so mild
When my poor heart you first beguiled
Why did ya run from me and the child
Johnny I hardly knew ya

[Chorus:]
We had guns and drums and drums and guns,
Hurroo Hurroo
We had guns and drums and drums and guns,
Hurroo Hurroo
We had guns and drums and drums and guns
The enemy never slew ya
Johnny I hardly knew ya

Where are the legs with which you run,
Hurroo Hurroo
Where are the legs with which you run,
Hurroo Hurroo
Where are the legs with which you run,
When first you went to carry a gun
Indeed your dancing days are done
Johnny I hardly knew ya

[Chorus]

You hadn't an arm, you hadn't a leg,
Hurroo Hurroo
You hadn't an arm, you hadn't a leg,
Hurroo Hurroo
You hadn't an arm, you hadn't a leg
You're a spineless, boneless, chickenless egg
You'll Have to be put with the bowl to beg
Johnny I hardly knew ya

[Chorus]

I'm happy for to see ya home,
Hurroo Hurroo
I'm happy for to see ya home,
Hurroo Hurroo
I'm happy for to see ya home
From the isle of Ceylon
Johnny I hardly knew ya

[Chorus]