They're actually similar in a lot of ways. For a defense of similarities see Stewart Kierkegaard's Relation to Hegel Reconsidered (https://www.amazon.com/Kierkegaards-Relations-Reconsidered-European-Philosophy/dp/0521039517). For more of the opposite, read Thulstrup's Kierkegaard's relation to Hegel.
Maybe to boil it down quite a bit,
- It's not clear how well Kierkegaard understood Hegel vs. Danish Hegelians.
- It's not agreed upon how important that distinction is.
- This leads to claims that Kierkegaard agrees with Hegel and disagrees with pseudo-Hegel.
Both positions strike me as extreme.
For the disagree on everything version, it's not historically plausible. In part this is because people in strongly similar cultures in roughly the same era are looking at the same questions. To put that another way, tribesman in the Amazon aren't considering whether or not it matters that someone takes a knee before a football game, but Americans apparently are. Or a more salient matter, is God an assumed feature of the discourse or something else? Kierkegaard and Hegel share a lot of these background assumptions.
Stewart to get to the they secretly agree on lots basically has to take passages where Kierkegaard says "Hegel is wrong" (paraphrasing here) and say that Kierkegaard isn't writing that about Hegel -- even when it's defensible as a valid reading of a Hegel passage that Kierkegaard is known to have owned.
To add my two cents on some relevant issues where they agreed,
- they believe to be a self is to be a relation.
- both think recognition and self-consciousness are important.
- both take a dialectical approach to their philosophy -- Kierkegaard with an ironic motivation at times and Hegel with much less irony.
- Both like Aristotle's work better than Plato (reflected in many similar arguments).
I could probably say a lot more.