Generally speaking, SMS and other messaging protocols are NOT HIPAA complaint. this is due to the inherent insecure nature of the protocols themselves. The data is in transit and its not encrypted. HIPAA law requires that all data in motion must be encrypted.
There are several HIPAA compliant texting applications on the market. See the requirements of a HIPAA compliant texting application here https://www.netsec.news/hipaa-compliant-texting-app/. These apps incorporate a range of security controls to ensure the confidentiality, integrity, and availability of ePHI and meet the requirements of the HIPAA Security Rule.
They incorporate access controls to ensure that only authorized individuals can gain access to messages containing ePHI. They maintain an audit trail so it is possible to see who has done what with ePHI. They have remote message deletion options to ensure that data is permanently erased in the event that a device is lost or stolen. The messages are also protected with end to end encryption.
TigerText is a good example of a HIPAA compliant messaging app.