The Hunter Ranger's Whirlwind Attack feature states:
You can use your action to make a melee attack against any number of creatures within 5 feet of you, with a separate attack roll for each target.
The rules on "Grappling" state (the rules in shoving are effectively identical):
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
I couldn't find any other features spells or abilities that allow a creature to make a flat "melee attack". If they had said "melee weapon attack" instead this would have meant the same thing but specifically have excluded grapple/shove as an option.
I am unsure whether the rules on grappling/shoving mean that the only time you can grapple/shove are when you take the Attack action, which you are not doing when using the Whirlwind Attack feature.
I would like answers to assume that the following ruling made in the Sage Advice Compendium is correct:
Whirlwind Attack is unusual, in that it’s a single attack with multiple attack rolls...