A good place to start is to note the other character that 一 is paired with, and determine if that character is being read with an on reading or a kun reading. Although there are exceptions, most two-character compound words are read as either on-on or kun-kun.
かぶ is a kun reading of 株, which makes it likely that the 一 in 一株 is also read with a kun reading.
Some counters can be read both in kun'yomi and on'yomi.
一度: いちど, ひとたび
一時: いちじ, ひととき
一足: いっそく, ひとあし
一色: いっしょく, ひといろ
In these cases, kun'yomi versions often have a specific meaning, and only 1, 2, or 3 can be used with these kun'yomi words (e.g. ひとたび, ふたたび or みたび are used, but よたび is rare, or ひととき is OK, but ふたとき is obsolete). Thus, these kun'yomi words are more likely a kind of compound word, rather than counter. Even so, the "on-on/kun-kun" rule is also applied.