I don’t know how special a Hungarianism it is, but very interesting indeed. This conjugation is a bit archaic and rural, you’ll barely meet it in texts from modern sources, but natives will understand it. Let’s start from the little bits, three verbs of the most common ones, all irregular of course. I show their roots as well.
eszik, e- – s/he eats
iszik, i- – s/he drinks
megy, me- – s/he goes
Note that these all third person singular forms, being that one the dictionary form.
Now let’s take the root and add the optative suffix -hat, -het which means “can, may, be able to, allowed to”. Iszik takes back vowel suffixes.
ehet – s/he can eat
ihat – s/he can drink
mehet – s/he can go
Now let’s add the conditional suffix -na, -ne which means “would.”
ehetne – s/he would be able to eat
ihatna – s/he would be able to drink
mehetne – s/he would be able to go
Let’s look around the personal suffixes for these forms.
ehetnék – I would be able to eat
ehetnél – you would be able to eat
ehetne – s/he would be able to eat
ehetnénk – we would be able to eat
ehetnétek – y’all would be able to eat
ehetnének – they would be able to eat
ihatnék – I would be able to drink1
ihatnál – you would be able to drink
ihatna – s/he would be able to drink
ihatnánk – we would be able to drink
ihatnátok – y’all would be able to drink
ihatnának – they would be able to drink
mehetnék – I would be able to go
mehetnél – you would be able to go
mehetne – s/he would be able to go
mehetnénk – we would be able to go
mehetnétek – y’all would be able to go
mehetnének – they would be able to go
Now, let’s see the first person singular forms: ehetnék, ihatnék, mehetnék. You can use them as phrasal words to urge someone you are depending on, and being slow they’ll delaying you. Ehetnék! “I would eat, it’s time to!” That’s not very polite, though. It’s nicer to use the third person plural and ask it as a question: Ehetnénk? “Could we eat, finally?”
But let’s stay with the first person singular. You can turn it into a noun. Just use it so, and use it in a possessive structure.
ehetnékem van – I have an Iwouldbeabletoeat
ihatnékom van – I have an Iwouldbeabletodrink
mehetnékem van – I have an Iwouldbeabletogo
This means, practically, “I’d like to eat, drink, go, I feel the urge to, it’s time for it”, but doesn’t contain the element of urging someone, you can use it when you’re preparing the food or drink or getting on the way yourself.