Yes, not only did Jesus himself also fast (Matthew 4:2), he also expected his disciples to do so (Matthew 6:16–18);(Matthew 17:21 NKJV: However, this kind does not go out except by prayer and fasting.) —fasts as a voluntary aid to faith and prayer—but the disciples of John were thinking about fasting in the Old Testament sense as an expression of sorrow and mourning at set times (Matthew 9:15; see also 1 Samuel 20:34; 2 Samuel 1:12; Daniel 10:2–3; Joel 2:12; Zechariah 7:3, Zechariah 7:5; Zechariah 8:19). The Mosaic law only mandated only one specific day of fasting: the Day of Atonement (Leviticus 16:29), but the Pharisees normally fasted on Mondays and Thursdays (see Luke 18:12), and John’s disciples did something similar—but Jesus’ disciples did not do so.
Yes, the core issue brought up by the disciples of John in Matthew 9:14 is not the fasting in and of itself, for not only did Jesus himself also fast (Matthew 4:2), he also expected his disciples to do so (Matthew 6:16–18; (Matthew 17:21 NKJV: However, this kind does not go out except by prayer and fasting.). Their fasts were personal, willing fasts as an aid to faith and prayer. What the disciples of John had in mind here, however, is fasting in the Old Testament sense as an expression of sorrow and mourning at set times (Matthew 9:15; see also 1 Samuel 20:34; 2 Samuel 1:12; Daniel 10:2–3; Joel 2:12; Zechariah 7:3, Zechariah 7:5; Zechariah 8:19). In the entire Mosaic Law, there is only one prescribed day of fasting: the Day of Atonement (Leviticus 16:29). The Jews, however, also had various other days of fasting. The Pharisees normally fasted twice every week (on a Monday and a Thursday, see Luke 18:12), and here we read that the disciples of John also participated in these types of fasts. Jesus and his disciples, however, did not.1
14 Then the disciples of John came to him, saying, “Why do we and the Pharisees fast, but your disciples do not fast?”