The description of the Barbarian Rage feature says:
Your anger lasts 1 minute. It ends early if you are stunned or stunned. if your turn ends and you have not attacked a hostile creature since your last turn or sustained damage since. You can also end your rage in turn as a bonus action.
To keep your rage from ending earlier, you must either attack a hostile creature or take damage between the end of your last turn and the end of your current turn.
As you point out, the boots are described as a "special melee attack":
By using the Attack action, you can create a special melee attack push a creature, either to hit her face down, or to push her away from you. If you can perform multiple attacks with the Attack action, this attack replaces one of them.
The grapple is described in the same way:
When you want to catch a creature or fight it, you can use the Attack action to create a special melee attack, a grapple. If you can perform multiple attacks with the Attack action, this attack replaces one of them.
Note that Rage does not say that the attack must "hit" – which can only happen with attack throws – which means it does not require any throws. # 39; attack. These special melee attacks are therefore a valid way to attack to keep raging. .
Rules designer Jeremy Crawford unofficially confirms it on Twitter, in response to some confusion about grapple / attack attacks, but not involving attack throws:
Does a barbarian keep his rage if he uses his action to attack if the attack is not an attack?
The Grapple option in the Player Handbook is an attack, but it uses a capacity check instead of an attack roll.
Here, he reiterates what is already indicated in the rules: the grappling hook and the crush are attacks. Attempting to grab or push a hostile creature allows you to continue your Rage.