The GameMastery Guide on Water says, "The rules presented in the Pathfinder RPG Core Rulebook for underwater combat apply to creatures not native to this dangerous environment [i.e. underwater], such as most PCs" (214). (Fortunately, monsters can be sorted by terrain.)

It seems reasonable that an eidolon that possesses gills be considered native to water, and, therefore, such an eidolon shouldn't suffer penalties when using its natural weapons during underwater combat, but you should check with your GM to be sure.

Also Paizo creative director James Jacobs in a 2012 post said, "Creatures that have the aquatic [sub]type don't have to worry about those [underwater combat] penalties[,] but all manufactured weapons must abide by these rules, regardless of whether or not the creature wielding them is aquatic." So, in conjunction with the GameMastery Guide's rule, if, for example, the eidolon opts for a greatsword instead its claws, the eidolon suffers the underwater combat penalties on attacks made with that weapon. (Note that Jacobs makes it clear that this is a ruling not a rule, but it seems okay and comes from an often authoritative source.)

About ranged attacks

Ranged weapons have a normal range and a long range. Under normal circumstances, you can attack enemies which are within your long range, but as soon as they are beyond normal range you have disadvantage on the attack. Attacking a creature while engaged in close combat does also impose disadvantage on ranged attacks. (PHB p.195)

Underwater Combat

In the special case of underwater combat, an attack beyond normal range is an automatic miss and an attack within normal range is at disadvantage unless it is one of the explicitly stated. This rule is a specific beats general case regarding ranged attacks and their two ranges.

[...] against a target within normal range, the attack roll has disadvantage unless the weapon is a crossbow, a net, or a weapon that is thrown like a javelin (including a spear, trident, or dart)

This means, that nets are excluded from the water's specific case rule regarding attacks within normal range. So the long range is a automatic miss rule still applies to them.

The disadvantage for attacks with a net is not caused by its range, but by close combat

The rules of ranged attacks in close combat do still apply. You could still use e.g. a crossbow under water to attack a creature 10ft away from you without getting disadvantage due to the range category, but when you are engaged in close combat you still get disadvantage on ranged attacks (unless you got the crossbow expert feat) same applies for nets.