The die from Spell Bombardment is a damage die, so it will be doubled on a critical hit
The section on "Critical Hits" states:
When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together. Then add any relevant modifiers as normal. To speed up play, you can roll all the damage dice at once.
Therefore, what we need to determine is whether Spell Bombardment's die counts as a damage die.
"Damage Dice" are never strictly defined by the rules, so they mean their natural English meaning, dice which are rolled that contribute to damage. This answer to the question "How do critical hits and bonus damage interact?" states:
Any damage expressed as dice gets to be rolled twice, but other modifiers are added only once.
Following this, any dice used in calculating damage are "Damage Dice" and so they are doubled on a critical hit.
The die from Spell Bombardment is a die roll that contributes to damage, thus it is a damage die, and so it is doubled on a critical hit.
There is also further support for this found in the Sage Advice Compendium:
Q. If a shadow rolls a critical hit, does it reduce the target’s Strength by 2d4, as well dealing the extra necrotic damage?
A. No. A critical hit lets you roll damage dice twice. An effect that deals damage is one that reduces the target’s hit points. The shadow’s Strength reduction isn’t damage, because it has no effect on the target’s hit points.
Note: Whether or not "negative dice modifiers" count as "damage dice" is discussed in the question "Does the Reduce option from the Enlarge/Reduce spell cause a critical hit to do 2d4 less damage?", but they do agree that "positive dice" count as damage dice.