The surname Milcak (in Czech or Slovak Milčák) falls into the class of less common surnames. Its origin and meaning is not possible to find in any Czech (or Slovak) word. During my searching in archives I found several its variations. Milcsak or also Milczak were the most common variants to current Milčák. I also discovered some records with Miltsak form.

Here are some illustrations of records from registry:
     
The question of origin and meaning can be answered thanks to the variation Milczak. Our surname is written in this form till now in Poland. Its derivation can be found from polish milczeć - be silent. From this verb is derived a noun milczek - a taciturn or mute person.

May it imply that the native Milcaks - "Taciturns" came from Poland?

Another variation of the surname is Milchak. I found it when I was searching for descendants of emigrants to the USA. Primarily I thought that this is different, only similarly sounding, surname. But finally it proved that most of the Milchaks have their ancestors in Slovakia and I also succeeded to find some links to Milcak family from Brutovce (see Milcaks from Brutovce).
I was thinking how this surname change could happened - quite simply, because of the sound of consonant "č". This sound was in the past written by means of ligatures cs, cz or ts. In English we can found this sound for example in words which or also witch. Thus the form Milchak is a variation which originates from the need to write somehow the sound "č".

In case of any difficulty with displaying the characters download this explanatory picture.

I am sorry to all linguist and phoneticists to whom my amateur explanation probably caused increase in heart rate and I welcome all comments and corrections.