The website you linked is unreliable and incorrect. I wouldn't recommend using it (or any other website) syllabification because syllabification is a highly controversial topic in linguistics. How to syllabify a word depends on who pronounces it. For example, the traditional pronunciation of the word film is [fɪlm] (one syllable), but I've certainly heard the disyllabic pronunciation [ˈfɪ.ləm].
The word 'dune' is usually pronounced [djuːn] and 'rider' [ˈɹaɪdə] (BrE). I agree with your analysis that 'dunerider' should be [ˈdjuːn.ɹaɪ.də] (three syllables) (it can also be pronounced in many other ways, though, even four-syllable pronunciation is possible depending on the speaker). It is not pronounced /ˈd(j)uː-n-raɪ-də/ (four syllables), because it suggests that the /n/ is syllabic which it is not. /n/ can only be syllabic when it's preceded by an obstruent (/t s z d/ etc) as in button. I would suggest you use your ears for deciding the number of syllables in a word.
Let's see how 'dunerider' is a three-syllable word:
This is the sonority curve for the word 'dunerider'. The dark blue dots mark the peaks of sonority and the number of peaks correspond to the number of syllables. As you can see, there are three peaks in [ˈdjuːn.ɹaɪ.də], hence three syllables.
The sonority of all phonemes of English can be depicted on a sonority scale (sonority hierarchy). A sonority hierarchy is a hierarchical ranking of speech sounds. Typical order of sonority values is:
Vowels [ɑ, ɔ, ɪ, i] etc > Glides [j, w] > Liquids [ɹ, l] > Nasals [m, n, ŋ] > Fricatives [s, f, θ, ð, z, ʃ] etc > Affricates [d͡ʒ, t͡ʃ]> Plosives [p, b, t, d, k, g]
Vowels are the most sonorous whereas plosives are the least sonorous sounds.
[From my previous answer]
I have explained the Onset Maximisation Principle (syllabification rule) in this answer.