As for the dependence on John on the LXX, we could note as one of many examples:
John 1:23: phóné boóntos en té erémó, euthunate tén hodon kuriou "The voice that cries in the wilderness, Make a straight way for the Lord"
Isaiah 40:3 LXX: phóné boóntos en té erémó, hetoimasate tén hodon kuriou, eutheias poieite tas tribous tou theou hémón "The voice that cries in the wilderness, Prepare the way for the Lord, make straight the paths of our God"
The only major difference is that the author of John has conflated the second and third clauses into one. The wordings of this verse in Aquila and Symmachus are quite different, illustrating the great variety of ways the Hebrew could be rendered into the Greek.
To respond to your specific comment, egó eimi is not just a "simple common Greek phrase that has nothing to do with the LXX". By itself, it just means "I am" but even here there are many different ways of saying this (e.g. just eimi with the subject understood, or just egó with the copular verb omitted, or in reversed order eimi ego); the context in which egó eimi occurs, however, can show that certain OT "I am" passages are related to certain Johannine expressions. As mentioned earlier, the use of the predicate absolute in subordinate clauses in John is strikingly like passages in Deutero-Isaiah. This pertains not just to the egó eimi itself but surrounding words in the OT text that occur in the Johannine texts:
"Be my witnesses as I am a witness, the Lord God says, and my servant whom I have chosen (exelexamén), so that men may know (gnóte) and believe (pisteuséte) and understand that I am [he] (hoti egó eimi)" (Isaiah 43:10; LXX).
"Unless you believe that I am [he] (pisteuséte hoti egó eimi), you will die in your sins....When you lift up the Son of Man, then you will know (gnósesthe) that I am [he] (hoti egó eimi)" (John 8:24, 28; cf. also John 13:19).
In this instance, it isn't just the common use of egó eimi, but the fact that (1) it is a predicate absolute, (2) it is preceded by the complementizer hoti, (3) and the subcategorizing verb of this dependent clause is either "believe" (pisteuséte in both) or "know" (both inflected forms of gnoó). Furthermore, in both cases the absolute use of egó eimi has theological or christological significance; the claim of egó eimi is to be an article of faith. Moreover, the conjunction of "believe" and "know" with a hoti-clause occurs elsewhere in John, again in a christological claim: "We believe (pepisteukamen) and know (egnókamen) that you are (hoti su ei) the Holy One of God" (6:69). Finally, a repeating cliche in Ezekiel is the phrase "you/they will know that I am the Lord (gnósesthe/gnóse hoti egó eimi kurios)" (kurios = YHWH in the Hebrew) in Ezekiel 7:9, 28:23, 24, 26, 29:6, 9, 16, 21, 30:8, 19, 25, 26, 32:15, 33:29, 34:27, 30, 35:4, 9, 12, 15, etc. The egó eimi is thus part of the allusiveness of the Johannine passages.
Note also that these two cases of absolute egó eimi in 8:24 and 8:28 precede and set up the absolute egó eimi in 8:58 ("Before Abraham was, I am"). Moreover, in 8:18 there is likely another allusion to Isaiah 43:10 LXX; in the former, Jesus calls himself a witness (egó eimi marturón), and in the latter (an interpolation in the LXX) God calls himself a witness (egó martus).
There are a number of other instances of absolute egó eimi in Deutero-Isaiah. One of these replaces the theological claim "I am Yahweh" in the Hebrew with the absolute egó eimi (Isaiah 45:18; LXX). Elsewhere there is the non-absolute egó eimi kurios or the doubled egó eimi egó eimi kurios "I am I am the Lord" or egó eimi ho theos "I am God" (Isaiah 45:8, 19, 22, 46:9, 48:17; compare 43:25). There is the use of the present tense absolute egó eimi to a non-present context (like in John 8:58) to indicate the changeless eternity of God: "From infancy to old age, I am (egó eimi), and until you have grown old, I am (egó eimi)" (Isaiah 46:3-4; LXX).
This is not to prove that John's use of egó eimi is itself based on the OT LXX, but there is some evidence of specific similarities. Moreover, egó eimi is not just a "simple common" phrase to John; it uses this phrase twice as much as all the other gospels combined, it has specific theological uses (cf. Koester's and Bultmann's treatment), and it appears to be a structuring feature of the gospel (cf. the "I am" discourses in 6:35, 8:12, 10:11, 11:25, 14:6, 15:1).