From the 2002 General Introduction to the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
“237. Then the principal celebrant, with hands joined, says the introduction to the Lord’s Prayer. Then, with hands extended, he says the prayer itself together with the other concelebrants, who also pray with hands extended and with the people.”
Clearly the priests are not permitted to hold hands. They are given explicit instructions to have their hands extended.
The deacon, altar servers and other ministers have these instructions from the Ceremonial of Bishps, Chapter 4, General Norms:
“
Joined hands
107 Unless the bishop is holding the pastoral staff, he keeps his hands joined [footnote 80: “Hands joined” means: “Holding the palms sideward and together before the breast, with the right thumb crossed over the left” (Caeremoniale Episcoporum, ed. 1886, I XIX, 1). when, vested, he walks in procession for the celebration of a liturgy; when he is kneeling at prayer; when he moves from altar to chair or from chair to altar; when the liturgical books prescribe joined hands.
**Similarly, concelebrants and ministers keep their hands joined when walking from place to place or when standing, unless they are holding something.”
(Ceremonial of Bishops, Liturgical Press, 1989, ISBN 0-8146-1818-9, page 43).
So this clearly makes it wrong for the altar servers to be holding hands.
The instructions are clear that it the priests who are to have there hands extended. If a deacon, instituted acolyte or altar server did it then he would be imitating the gesture proper to the priest. This is condemned in the 1997 Instruction On Certain Questions, Article 6 §2:
“… Neither may deacons or non-ordained members of the faithful use gestures or actions which are proper to the same priest celebrant. …”.
(The full document is at
vatican.va/roman_curia/pontifical_councils/laity/documents/rc_con_interdic_doc_15081997_en.html ).
The 1963 Constitution on the Liturgy Sacramentum Caritatis has:
“31. The revision of the liturgical books must carefully attend to the provision of rubrics also for the people’s parts.”
(The full document is at
vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19631204_sacrosanctum-concilium_en.html ).
So if it wanted to the people to hold hands for the Our Father it would say so. They are deliberately left out of the instructions to have hands extended like the priests concelebrating. To adopt another, similar gesture of holding hands with people next to them is not correct.