No, he should not. The breaking of the host is known as the "fraction," and there is a special place for it in the Mass—namely, in the Fraction Rite, which occurs after the Sign of Peace and immediately before the Communion Rite.
Since the Church has a specific place in the liturgy for the fraction, to perform it at another time subverts the role of the Fraction Rite and must not be done.
Further, the rubrics in the Sacramentary tie the meaning of the Fraction Rite to the commingling, where a piece of the host is placed in the chalice. The symbolism of this is commonly explained today as representing the resurrection of Christ, the reuniting of his Body and Blood.
The rubrics of the Mass link the meaning of the fraction to the commingling, stating: "Meanwhile, [the priest] takes the host and breaks it over the paten. He places a small piece in the chalice, saying inaudibly: ‘May this mingling of the Body and Blood of our Lord Jesus Christ bring eternal life to us who receive it.’"
Since nothing else is said—either in the rubrics or the prayers—about the breaking of the host, its primary purpose in the current order of Mass seems to be to obtain a piece of the host for use in the commingling. Any other meaning attached to the fraction that precedes the commingling would be secondary.
If one breaks the host on the words "He broke the bread," it would have a different primary meaning—either a reference Jesus’ breaking the bread for his disciples to partake or to the breaking of his body on the cross or both. Thus it would amount to adding a new rite to the Mass, which cannot be done (see below).
Snapping a host in two on the word "broke" is also dangerous. It is done so quickly and carelessly that excessive particles are likely to result and possibly be scattered. Priests who do it may think that they are heightening the symbolism of the Mass, but they are actually detracting from it as well as giving scandal to many of the faithful.
As always, the dictum of Vatican II applies: "No person, even if he be a priest, may add, remove, or change anything in the Liturgy on his own authority" (Sacrosanctum Concilium 22, cf. CIC 846 §1).