You are correct that Teradata JDBC Driver 13.10 and earlier versions are not supported for use with JDK/JRE 7.
Teradata JDBC Driver 14.0 or 14.10 are supported for use with JDK/JRE 7, and Yes, we recommend that you upgrade your Teradata JDBC Driver first before upgrading your JRE version. Teradata JDBC Driver 14.0 and 14.10 support JDK/JRE 1.4.2 through 7, so unless you are still using some really old JRE 1.4.0 or 1.4.1 versions, you will be OK.
Our Teradata JDBC Driver supported platform matrix is available here on Developer Exchange:
http://developer.teradata.com/doc/connectivity/jdbc/reference/current/platformMatrix.html
You are correct that Teradata JDBC Driver 13.10 and earlier versions are not supported for use with JDK/JRE 7.
Teradata JDBC Driver 14.0 or 14.10 are supported for use with JDK/JRE 7, and Yes, we recommend that you upgrade your Teradata JDBC Driver first before upgrading your JRE version. Teradata JDBC Driver 14.0 and 14.10 support JDK/JRE 1.4.2 through 7, so unless you are still using some really old JRE 1.4.0 or 1.4.1 versions, you will be OK.