I will answer in terms of Grammar only, since I have very limited knowledge in theology.
The present subjunctive phrase is "so that he has eternal life" ( ἵνα ἔχῃ ζωὴν αἰώνιον) and not just "have eternal life".
"ἵνα ἔχῃ ζωὴν αἰώνιον" is dependant upon "τὸν Υἱὸν τὸν μονογενῆ ἔδωκεν" and not upon the word "πιστεύων" (= the person who believes)
[The word ἵνα (=so that) is always combined with subjunctive and it indicates the purpose why something happens. The condition upon which this purpose is dependant, is indicated by the main clause. In this context the main clause is "τὸν Υἱὸν τὸν μονογενῆ ἔδωκεν". (= Gave His only Begotten Son)]
"πιστεύων"(= the person who believes) is the subject of the verb "ἵνα ἔχῃ" (and also subject of the verb "ἵνα μὴ ἀπόληται")
"πιστεύων" is a continuous form, so it means the person who believes constantly.
"ἵνα ἔχῃ" is present subjunctive, so it is a continuous form as well. It indicates that the person will constantly have an eternal life.
Therefore, the grammar and the syntax of the specific phrase indicates that the person who believes constantly, will (constantly) have an eternal life.
If someone stops believing, I don't know in theological terms what may happen. It is not mentioned in the phrase.