Neither Harry nor Barty Crouch Jr knew about the eggs when preparing for the First Task, so Summoning the eggs never occurred to them.
Many of the answers here are premised on wiki-based speculation. There may have been anti-Summoning charms on the eggs. There may not have been. We simply do not know either way. What we do is that the very existence of the eggs and the precise nature of the First Task was a closely guarded secret. The existence of the dragons obviously leaked to the champions before the task but the existence of the eggs seems not to have been leaked. All that the champions knew is that they had to somehow "get past" the dragons. The part about capturing the eggs was only revealed just before the task itself; until then the egg element of the Task was a mystery.
“Well, now we’re all here - time to fill you in!” said Bagman brightly. “When the audience has assembled, I’m going to be offering each of you this bag” - he held up a small sack of purple silk and shook it at them - “from which you will each select a small model of the thing you are about to face! There are different - er - varieties, you see. And I have to tell you something else too...ah, yes...your task is to collect the golden egg!”
(Goblet of Fire, Chapter 20, The First Task).
Charlie Weasley was part of the team which was responsible for looking after the dragons and not even he knew what the First Task was or why the eggs were required.
“Four...” said Hagrid, “so it’s one fer each o’ the champions, is it? What’ve they gotta do - fight ’em?”
“Just get past them, I think,” said Charlie. “We’ll be on hand if it gets nasty, Extinguishing Spells at the ready. They wanted nesting mothers, I don’t know why..."
(Goblet of Fire, Chapter 19, The Hungarian Horntail).
It's true that Harry could potentially have tried Summoning the egg on impulse after he'd found out about them. However, his objective was to have a plan in advance for getting past the dragon; changing his plan at the last minute would've been risky. Crouch's whole plan was to guide Harry through the tasks by preparing him as best as possible. Regardless of the approach Harry was to take in the task, it had to be agreed in advance. That way, Harry could feel confident that he just had to execute the plan which he has in his head. Changing the plan to bypass the broom may have simplified it but would have made Harry less secure in what he was trying to do. It would've meant Harry having to think on his feet - which may be what the Task was actually trying to test but which was specifically was Crouch was trying to avoid. As it was, Harry was much more confident and assured when using the broom.
He swung his leg over the broom and kicked off from the ground. And a second later, something miraculous happened...
As he soared upward, as the wind rushed through his hair, as the crowd’s faces became mere flesh-colored pinpricks below, and the Horntail shrank to the size of a dog, he realized that he had left not only the ground behind, but also his fear...He was back where he belonged...
(Goblet of Fire, Chapter 20, The First Task).
So neither Crouch nor Harry knew about the eggs and made a plan based on using the Firebolt. Deviating from that plan would not have been a good idea.
However, even if they had known about the eggs in advance they may well have decided to use the Firebolt anyway. The Firebolt gave Harry a significant defensive advantage. Whilst airborne he could avoid the Horntail and stay a safe distance away from it. There's no point in Summoning the egg if you're immediately consumed by a burst of fire.
By the by, if there wasn't an Anti-Summoning Charm on the eggs then Harry could've Summoned it. The location of the Horntail would've affected this. She was sitting next to the eggs, not on them.
He looked down at the clutch of eggs and spotted the gold one, gleaming against its cement-coloured fellows, residing safely between the dragon's front legs.
(Goblet of Fire, Chapter 20, The First Task).
There's also nothing about the Summoning Charm which means you have to own the item you're summoning.