All bugs are indeed insects (members of the order Hemiptera); all insects, however are not bugs, if one uses the term in a precise, scientific way. If one uses the term in a general, informal way to refer to any small, terrestrial invertebrate, then presumably all insects are bugs, too. Either way, the sentence doesn't make much sense!
I don't know what exactly the Hungarian word 'bogár' means, however. The dictionary I have gives both 'insect' and 'beetle', whereas the Hungarian Wikipedia article on bogarak seems to use the term to refer specifically to beetles (Coleoptera). Can a Hungarian speaker perhaps clarify the difference between rovarok and bogarak?
bogár = bug, beetle
"Beetles are a group of insects that form the order Coleoptera. The word coleoptera is from the Greek κολεός, koleós, meaning "sheath"; and πτερόν, pterón, meaning "wing", thus "sheathed wing", because most beetles have two pairs of wings, the front pair, the "elytra", being hardened and thickened into a shell-like protection for the rear pair and the beetle's abdomen."
rovar = insect (more formal (and rather used in science topics) word for the same is 'ízeltlábú'
"Insects (from Latin insectum, a calque of Greek ἔντομον [éntomon], "cut into sections") are a class of invertebrates within the arthropod phylum that have a chitinous exoskeleton, a three-part body (head, thorax and abdomen), three pairs of jointed legs, compound eyes and one pair of antennae."
ízeltlábú comes from 1. the verb of ízel, ízekre szed = cut into parts, smaller sections 2. the noun of láb = leg
all in all, all bugs are insects, but not all insects are bugs
I have often heard people restrict the word "bug" to insects, though not only hemiptera (my daughter was very into entomology when she was little, and insisted on that distinction). More broadly, I have heard it used of all arthropods, even shrimp, lobsters, etc, or even all small organisms, such as bacteria or copepods. In Hungarian, I have heard "bogár" used most specifically of beetles, such as the katicabogár, the ladybird beetle, but have also heard it used indeterminately of little, crawly things.
Many bugs are not insects but other small arthropods: gömbászka (pill bugs), for example. That is why the scientifically delineated group is called the true bugs.
Mostly all insects are beetles -- God really loves beetles (or the Beatles, to the neglect of the Crickets they took their name from) as they say. Flies are descended through ants, bees, and wasps from beetles. Still I'd like to hear about the etymology of the word bogár, as this sort of science is too recent. Surely it meant something different in the past? People these days, they think words are just about science -- why do they think Linnaeus gave us scientific names?