Illusory Gains — ruling 1

Illusory Gains's last ability is mandatory. You can't choose to have Illusory Gains not move.

Illusory Gains — ruling 2

If Illusory Gains can't legally enchant the creature that enters the battlefield (perhaps because it has protection from blue), it remains where it is. You retain control of the creature it's currently enchanting.

Illusory Gains — ruling 3

The last ability of Illusory Gains doesn't target the new creature. It will become attached to that creature even if that creature has hexproof or shroud.

Illusory Gains — ruling 4

If multiple creatures enter the battlefield under an opponent's control at the same time, Illusory Gains will trigger for each of them. You may put these abilities on the stack in any order. As each resolves, you'll briefly gain control of the associated creature. The last one to resolve determines which one you'll ultimately control.

Illusory Gains : règles, fonctionnement et interactions (MTG)

Illusory Gains is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

Enchant creature You control enchanted creature. Whenever a creature an opponent controls enters, attach this Aura to that creature.

How it works : Illusory Gains

Original English text — Wizards of the Coast / Scryfall

#1February 25, 2015· Wizards of the Coast
Illusory Gains's last ability is mandatory. You can't choose to have Illusory Gains not move.
No other card is named in this ruling.
#2February 25, 2015· Wizards of the Coast
If Illusory Gains can't legally enchant the creature that enters the battlefield (perhaps because it has protection from blue), it remains where it is. You retain control of the creature it's currently enchanting.
No other card is named in this ruling.
#3February 25, 2015· Wizards of the Coast
The last ability of Illusory Gains doesn't target the new creature. It will become attached to that creature even if that creature has hexproof or shroud.
No other card is named in this ruling.
#4February 25, 2015· Wizards of the Coast
If multiple creatures enter the battlefield under an opponent's control at the same time, Illusory Gains will trigger for each of them. You may put these abilities on the stack in any order. As each resolves, you'll briefly gain control of the associated creature. The last one to resolve determines which one you'll ultimately control.
No other card is named in this ruling.