0344 numbers are not free to call on Vodafone tariffs, nor are they free from any other network provider.
0344 numbers are national telephone helplines which are charged at the same rate as geographic 01 and 02 numbers, which means you will be able to use your Vodafone weekend bundle and inclusive minutes to call these numbers for free. However you will still lose your minutes when phoning these helplines, alternatively if you do not have any remaining minutes allocated on your tariff you will have to pay an access charge of 19p on landlines or up to 55p on mobiles plus a per-minute fee outline in your Vodafone contract.
A brief background on 0344 numbers
0344 numbers were introduced in the UK as early as 2007, however their popularity only increased dramatically when Ofcom regulations were changed in 2015. This change in policy meant that more expensive service numbers which start in 084 and 087 prefixes are no longer allowed to be used as anything other than sales hotlines. Several companies simply changed their existing 0844 numbers to the cheaper 0344 prefix to help their customers to convert to the new system.