The Eighth Day (or Last Great Day) is a separate festival from the Feast of Tabernacles, which can only derive its significance in the New Testament.
If we do not keep God's holy days, we will deprive ourselves of the knowledge of God's purpose. Jesus and the first century church observed and upheld these days.
John chose to highlight the healing of a crippled man at the Pool of Bethesda. The pool, the man healed, and Christ's curious question are all significant.