What is the zen influence behind this Basho poem My eyes following until the bird was lost at sea found a small island?

I believe you are referring to:

ほととぎす消え行く方や島一つ

hototogisu kieyuku kata ya shima hitotsu.

(summer) little cuckoo. it disappears. a single island

The "Zen influence" can be found on two levels:

1. It is not so much in the Words themselves, but in the experience it describes.

2. Buddhist terms with the same pronunciation as some key Words in the poem.

#1 is impossible to explain, it must be experienced.

As for #2:

hototogisu is a "season-Word", something that is included in almost all haiku and hokku (this poem is a hokku). This particular season Word is a species of bird, but indirectly refers to the summer.

ho is also the pronunciation for the Buddhist Teachings, 法

toto also means "attainer of perfection" 度得

gisu also means "pillar", or "center", å®—

shima also means "that which is derived", i.e. interpreted/analyzed, 枝末

You could therefore also hear the poem as:

Attaining the perfect core of the Buddha's Teachings - it disappears - all separation becomes one.

The poem itself is a reference to a much older poem found in the Kokinshū (#409).