In the first place, be sure to use a version of Connector/J no older than v5.1.37, where support for the JSON datatype was added, and preferrably no older than v5.1.40, which fixed some JSON-related bugs. But if that's an issue for you, then it is at best secondary.
Before that even matters, JdbcTemplate needs to understand your JSONObject argument. The particular method you are using documents that the variable arguments, including your JSONObject are
arguments to bind to the query (leaving it to the PreparedStatement to
guess the corresponding SQL type); may also contain SqlParameterValue
objects which indicate not only the argument value but also the SQL
type and optionally the scale
When it says the PreparedStatement will guess, it can only mean the two-arg version of PreparedStatement.setObject() which documents:
The JDBC specification specifies a standard mapping from Java Object
types to SQL types. The given argument will be converted to the
corresponding SQL type before being sent to the database.
Note that this method may be used to pass datatabase- specific
abstract data types, by using a driver-specific Java type. If the
object is of a class implementing the interface SQLData, the JDBC
driver should call the method SQLData.writeSQL to write it to the SQL
data stream. If, on the other hand, the object is of a class
implementing Ref, Blob, Clob, NClob, Struct, java.net.URL, RowId,
SQLXML or Array, the driver should pass it to the database as a value
of the corresponding SQL type.
(Emphasis added.)
JDBC does not have built-in support for a JSON datatype. The docs do allow for Connector/J to provide a driver-specific type corresponding to the JSON SQL datatype, but just because MySQL and Connector/J support JSON does not mean you can pull a random JSON-representing object off the shelf and present it to them, nor indeed that they provide or accommodate a JSON-specific Java datatype at all. Inasmuch as MySQL seems not to publish or reference any documentation of JDBC extension APIs in its Connector/J developer guide, I'm inclined to doubt that it recognizes any such type.
It seems, then, that you should rely on the same technique that you would do when programming directly in MySQL's SQL dialect:
In MySQL, JSON values are written as strings. MySQL parses any string
used in a context that requires a JSON value, and produces an error if
it is not valid as JSON. These contexts include inserting a value into
a column that has the JSON data type and passing an argument to a
function that expects a JSON value [...].
(MySQL 5.7 Reference Manual, section 11.6)
That is, convert your JSONObject to a JSON-format String, and present it to your template that way. On retrieval, expect to have to do the reverse.